Inverkan av t ¨
ojnings och
punkt-defekter p ˚a Seebeck-koefficienten
f ¨
or termoelektrisk CoSb
3
Influence of strain and point defects on the
See-beck coefficient of thermoelectric CoSb
3Examensarbete, 30 hp, Computational Materials Science VT 2021
Sammanfattning
M˚anga studier och experiment har genomf¨orts under ˚aren f¨or att hitta l¨osningar till de nuvarande utmaningarna med elektriciteten. Problematiken ¨
ar inte enbart relaterat till hur br¨anslen f¨orbrukas men ¨aven till milj¨ofr˚agor som har vuxit st¨orre och utvecklat sig till ett allvarligt problem som har lett till milj¨of¨ororeningar och ozonskador. D¨arf¨or har behovet f¨or alterna-tiva energik¨allor blivit ett av de fr¨amsta m˚alen och forskningen har b¨orjat att syssels¨atta sig mer och mer med termoelektrisk. Detta ¨ar ett tydligt tillv¨agag˚angss¨att f¨or att producera ny energi genom att omvandla v¨arme direkt till elektricitet.
Koboltantimonid (CoSb3) tillh¨or en bred grupp av material med
skutteru-ditstruktur som nyligen har identifierats som potentiella nya termoelek-triska material med h¨og prestanda. CoSb3 ¨ar en av flera l¨oftesrika
ter-moelektriska material med en intermedi¨ar temperaturvariation. Det bin¨ara CoSb3 ¨ar en halvledare med ett smalt bandgap och en relativt platt
band-struktur och en utm¨arkt elektrisk prestanda. Verkningsgraden av den ter-moelektriska prestandan av bin¨ar koboltantimonid m¨ats genom sitt god-hetstal. Godhetstalet ¨ar viktigt f¨or termoelektriska material och styrs framf¨orallt av Seebeck-koefficienten f¨or att den tydligg¨or en nominell be-lastningsfaktor. Seebeck-koefficienten av CoSb3 kan p˚averkas genom m˚anga
faktorer som antingen ¨okar eller minskar koefficienten. P˚ak¨anningar ¨ar en viktig del av transportegenskaperna d¨aribland Seebeck-koefficienten. M˚alet med den h¨ar studien var att unders¨oka effekten av punktdefekter och p˚ak¨anningar p˚a Seebeck-koefficienten av skutteruditen CoSb3.
Den bin¨ara CoSb3 skutteruditen unders¨oktes med hj¨alp av density
func-tional theory (DFT) f¨or att kunna ber¨akna marktillst˚andsegenskaperna, framf¨orallt Seebeck-koefficienten. Det togs h¨ansyn till tv˚a olika CoSb3
strukturer, en ideal utan defekter och en annan som betecknas som real, dvs. att den inneh˚aller defekter. I b˚ada fall unders¨oktes Seebeck-koefficienten och dess svar medan p˚ak¨anningar appliceradess genom att ¨andra struk-turvolymen. Non-equilibrium Green’s funktionen anv¨ands som j¨amvikt inom en DFT-simulering f¨or att f˚a en en f¨ordelning av ¨overf¨oringen som f¨oruts¨atter en ber¨akning av Seebeck-koefficienten. Dessutom placerades syremolekyler p˚a (001) ytan av en 2x2x1 CoSb3 supercell f¨or att se om en
oxidering leder till att en punktdefekt bildas. Dessa simuleringar utf¨ordess med hj¨alp av en DFT–baserad molekyldynamik.
ide-ala struktur. Vid sammantryckning ¨okade det totala v¨ardet av Seebeck-koefficienten. D¨aremot bytte Seebeck-koefficienten fr˚an negativ till positiv och ¨okade till 894 µVK−1 under belastning, n˚agot som var ov¨antat. En kartl¨aggning av hur elektront¨atheten f¨ordelas unders¨oktes f¨or att kunna f¨orklara reaktionen av Seebeck-koefficienten p˚a j¨amvikt, sammantryckning och belastning. Det uppt¨acktes att f¨ordelningen av elektronerna mellan Co och Sb hade h¨ojts under sammantryckning vilket betyder en ¨okat ¨ overlapps-matris (covalent interaction). D¨aremot s¨ankte belastningen f¨ordelningen av elektronerna mellan Sb och Co.
Den reala strukturen f¨ororsakats av oxidering visade tomrum f¨or Sb. Seebeck-koefficienten p˚averkades annorlunda ¨an den med en ideal struktur. Un-der j¨amvikten ¨okade Seebeck-koefficienten till 151 µVK−1. F¨ordelningen av elektront¨atheten mellan Sb och Co f¨orb¨attrades i den reala strukturen j¨amf¨ort med den ideala strukturen. Den mest drastiska f¨or¨andringen uppt¨ ack-tes under belastningen d˚a Seebeck-koefficienten n˚adde −270 µVK−1. Det kan diskuteras om detta uppst˚ar p˚a grund av O som ¨okar ¨overlapsmatrisen. Metoden som introducerades i denna studie ¨ar en samverkan av defekter och belastningseffekter som kan vara f¨ordelaktiga f¨or andra termoelektriska material.
Abstract
Many studies and experiments have been conducted over the years to find solutions to the electricity problem. This issue is not just related to how fossil fuels are dispensed. Also, the environmental concerns associated with using fossil fuels have become a severe issue, which is a major cause of envi-ronmental pollution and ozone layer damage. As such, the need for energy becomes one of the essential goals. Therefore, research has begun to revolve around thermoelectrics, which is a straightforward approach for generating energy, by converting heat directly into electricity.
Cobalt antimonide (CoSb3) belongs to a broad family of materials with
the skutterudite structure, which have been recently identified as potential new thermoelectric materials with high performance. The CoSb3 is one
of the numerous promising thermoelectric materials in the intermediate temperature range. The binary CoSb3 is a narrow bandgap semiconductor
with a relatively flat band structure and excellent electrical performance. The thermoelectric performance efficiency of binary CoSb3 is measured by
its figure of merit. The figure of merit is important for thermoelectric materials and is primarily governed by the Seebeck coefficient because it exhibits a square dependence. The Seebeck coefficient of the CoSb3 can
be affected by many factors that can either increase or decrease it. Strain is an important aspect for the transport properties, including the Seebeck coefficient. The goal of this thesis project is to study the effect of point defects and strain on the Seebeck coefficient of skutterudite CoSb3.
The binary CoSb3 skutterudite was explored through density functional
theory (DFT) to calculate the ground-state properties, in particular the Seebeck coefficient. Two different CoSb3 structures were considered, an
ideal one (without any defects) and the other was termed real (containing defects). In both cases, the Seebeck coefficient and its response were studied while strain was applied by changing the volume of the structure. The non-equilibrium Green’s function was used within a DFT simulation to get a transmission distribution, where it was essential for calculating the Seebeck coefficient. Moreover, oxygen molecules were placed over the (001) surface of 2 × 2 × 1 CoSb3 supercell to establish if oxidation leads to point defect
formation. These simulations were carried out by DFT-based molecular dynamics.
It is found that the strain affects the Seebeck coefficient in the ideal struc-ture. At compression, the absolute value of the Seebeck coefficient
in-creases. By contrast, the Seebeck coefficient changed its sign from negative to positive and increased to 894µVK−1 at tension, which was unexpected. The electron density distribution map was explored to explain the behavior of the Seebeck coefficient at equilibrium, compression, and tension. It can be seen that the electron distribution between Co and Sb is increased at compression, implying an increased orbital overlap (covalent interaction). By contrast, the tension reduces the electron distribution between Sb and Co.
The real structure induced by oxidation exhibits Sb vacancies. The See-beck coefficient is affected differently than that of the ideal structure. At equilibrium, the Seebeck coefficient increases to 151 µVK−1. The electron density distribution between Sb and Co is enhanced in the real structure compared to the ideal one. The most drastic change is found at tension, where the Seebeck coefficient reaches −270 µVK−1. It may be speculated that this occurs due to O which increases the orbital overlap. The strategy introduced in this work, an interplay of defects and strain effects, may be beneficial for other thermoelectric materials.
Contents
1 Introduction 1
2 Methods 3
2.1 Density Functional Theory . . . 3
2.2 Equation of State . . . 5
2.3 The Boltzmann Transport Equation . . . 6
2.4 Non-Equilibrium Green’s Functions . . . 7
2.5 The Seebeck Coefficient . . . 8
2.6 Density Functional Theory Based Molecular Dynamics . . . 9
3 Results and Discussion 11 3.1 Ideal CoSb3 . . . 11
3.2 Real CoSb3 . . . 16
4 Conclusions 25
5 Future Work 26
1
Introduction
The binary cobalt antimonide (CoSb3) is one of the most important system
used for thermoelectric (TE) applications and it belongs to the skutterudite family [1]. These compounds are distinguished by many characteristics. They are semiconductors with a cubic structure belonging to the space group 204 (Im-3). CoSb3 is characterized by a small bandgap of
approxi-mately 0.16 eV [1]. These compounds have the composition MX3, where M
is a transition metal, such as Co, which has the Wyckoff symmetry coor-dinates at the 8c (1/4, 1/4, 1/4) site, and X represents a pnictogen atom, such as Sb, at the 24g (0, y, z) Wyckoff site with the internal parameters of y = 0.332 and z = 0.157 as well as lattice parameter a = 9.0347 ˚A [2]. These compounds are composed of 32 atoms and contain two voids per unit cell which allows for external atoms to enter the structure.
A schematics of the unit cell structure is shown in figure 1. CoSb3 has a
broad photoluminescence band with maximum intensity at 409 nm as no-ticed for nanoparticles synthesized with sodium dodecyl sulphate [3]. The binary skutterudites could exhibit either p- or n-type conduction. In the Sb-rich compositions a p-type conduction is observed, while in the Co-rich compositions there is a transition from n- to p-type conduction above 500 K [4].
TE performance efficiency of binary cobalt antimonide is measured by its figure of merit (ZT )
ZT = α
2T
ρk (1)
where α is the Seebeck coefficient, T is absolute temperature, ρ designates electrical resistivity, and k is thermal conductivity. Therefore, efficient TE materials must have large α, small ρ, and small k. α goes with the square so it is obviously the most important parameter in the figure of merit. Therefore, the current study is focused on the Seebeck coefficient.
Figure 1: The unit cell of CoSb3, where the transition atoms (blue) form a
cubic sublattice and the pnictogen atoms (brown) are arranged to form 6 planes and keep two voids empty.
Strain is an important aspect for the transport properties, including the Seebeck coefficient, since it has been reported for CoSb3 under strain that
the pressure enlarges the bandgap from 0.13 to 0.42 eV resulting in the en-hanced Seebeck coefficient [5]. Another important aspect is the influence of point defects on the transport properties, since it has been reported that the growth induced point defects affect the thermal transport (excess Sb in CoSb3) [7]. This was for a sample with 6 at.% excess of Sb, which showed
an intrinsic positive Seebeck coefficient of 200 µVK−1 and enhanced the ZT value, reaching a maximum of about 0.1 at 350 °C, being two times higher than the pristine (ideal) CoSb3. The real structure (configurations
with defects are termed here as real) shows a negative Seebeck coefficient of −430µVK−1 at room temperature [7]. If a point defect is an Sb vacancy, a significant impact on the bandgap occurs and hence it alters on the See-beck coefficient [7]. Besides growth induced point defects, other origins thereof are possible, such as oxidation, but this has not yet been explored. However, it is known that TE materials oxidize at elevated temperatures [8].
Many questions still remain open. What kind of point defects induced by oxidation can form? Would oxidation give rise to exactly the same defects as mechanical loading or growth? Could such defects affect the transport properties in the same way or not? Typically, when a material is oxidizing there is a mass transport and the oxygen atoms may enter the structure and some constituting elements may leave the structure. Such mass trans-port may leave vacant sites [16]. The working hypothesis is that oxidation induced defects are relevant for the transport properties of CoSb3.
2
Methods
2.1
Density Functional Theory
Each material is characterized by different structures and properties, mak-ing scientists search for different theories to study the materials and their properties. Among these theories is quantum mechanics, which provides a description of the physical properties of nature at the scale of atoms and subatomic particles. That can be evident from the knowledge and under-standing reached through the growth of technology applications that would not have been achieved without quantum mechanics.
The Schr¨odinger and Dirac equations are used within the concept of quan-tum mechanics [6]. However, it is difficult to apply these equations to macroscopic systems of the order of 1023. Then an alternative is found, using a theory called density functional theory (DFT). The main idea be-hind DFT is that the energy (or other properties) of a system can be expressed as a function of electron density. This electron density is defined as the probability of finding one electron in an infinitesimal volume. This function is represented by only three spatial coordinates, and the integra-tion over all the space yields the number of electrons of the system [6]. The ground-state electron density and the total energy could be calculated within the Kohn Sham (KS) formulation of DFT [18], which assumes that the density of a system interacting electrons can be obtained as the den-sity auxiliary system of non-interacting electrons that move in an effective density-dependent potential [19].
DFT allows parameter-free calculations of densities, ground-state energies, and related quantities such as lattice structure and constant, phonon des-peration relation, elastic constants, and magnetic moments, etc [14]. DFT uses electron density to describe the intricate many-body effects within a single particle formalism, usually very useful approximations to the physi-cal fields.
DFT calculations were performed for the ideal CoSb3 structure using the
OpenMX simulation package [31]. Computations were implemented us-ing the Perdew-Burke-Ernzerh of (PBE) exchange correlation functional
[30], within generalized-gradient-approximation (GGA) [20]. The norm-conserving-potentials (VPO) [34] were generated with the pseudo-atomic-orbitals (PAO) [33] approaches for Sb and Co to provide more ac-curate description of the CoSb3. The basis functions were generated within
a confinement scheme [32] and designated as: Co6.0S-s2p3d2f1 and Sb7.0-s3p3d3f2 (Co and Sb denote the chemical element for CoSb3, followed by
the cutoff value in Bohr radii and the last set of characters are the primi-tive orbital for each element). A skutterudite in ideal CoSb3 structure was
studied by implementing 32 atoms, including 8 Co atoms and 24 Sb atoms.
The structural relaxation was done at 300 K where the lattice parameter was 9.0347 ˚A (as an experimental value) and a kinetic energy cutoff was 150 Ry. The K-grid integration was handled carefully to ensure the numerical convergence where the Brillouin zone (BZ) [13] was sampled at K-grid 5 × 5× 5 for the lattice parameter, bulk modulus, and the Seebeck coefficient. The calculations were repeated for K-grid 7 × 7 × 7, and K-grid 11 × 11 × 11 to ensure numerical convergence. Because the computation was precised at K-grid 7× 7× 7, all the DFT calculations were performed at that K-grid.
In this section, DFT was also used to investigate the effect of point defect on the Seebeck coefficient in real CoSb3. The real CoSb3 structure was
studied by implementing 31 atoms, including 8 Co, 22 Sb atoms, and one oxygen atom. The point defect was introduced by removing two Sb atoms from the CoSb3 structure. This is due to the oxidation that leads to
vacan-cies in the CoSb3 structure. The vacancies of Sb and added oxygen atom in
the structure were chosen randomly from the simulation of oxidation. The norm-conserving VPO were generated together with the PAO approaches for Sb, Co, and O atoms. The basis functions were generated within a con-finement scheme by added oxygen atom were designated as: O5.0-s2p2d1 (O denotes the chemical element, followed by the cutoff value in Bohr radii and the last set of characters are the primitive orbital for each element). The temperature, BZ, and the kinetic energy cutoff were implemented sim-ilarly to the ideal structure. In order to be able to compare the results of the Seebeck coefficient in real and ideal CoSb3, the structure relaxation
2.2
Equation of State
Equation of state (EOS) is an analytical expression relating pressure to the volume and temperature. The isothermal EOS represents the relation between pressure and volume at a constant temperature, while the relation between the volume and temperature at constant pressure is known as the isobaric EOS. However, the isochoric EOS relates temperature and pressure at constant volume [10]. The EOS has been derived by many authors based on different physical assumptions. Among these assumptions, the equation of state depends on finite-strain theory [10]. This theory has played a significant role in developing EOS for solid. The Birch–Murnaghan EOS based on the Eulerian strain theory has been widely used for understanding the high-pressure behavior of solids [11]. The Birch–Murnaghan defined as:
E(V )=E0+9V160B0 h V0 V 2/3 − 1i3B00 +h V0 V 2/3 − 1i2h6 − 4 V0 V 2/3i (2) P (V )=(3 B0/2) h V0 V 7/3 − V0 V 5/3i ×n1 + 0.75 (B00 − 4)h V0 V 2/3 − 1io(3)
here, E is the total energy, V is the volume, and E0 presents the minimum
energy, V0 the volume corresponding to that energy, B0 is the bulk
modu-lus, P is the pressure and B00 represents the bulk modulus derivative with respect to the pressure.
After the structural relaxation, the unit cell of CoSb3 was used to
calcu-late the mechanical properties by performing seven finite distortions of the lattice parameter. The total electronic energy of the primitive cell volumes ranging from 681.47 to 857.37 ˚A3 was calculated. The dependence of cell volume on total electronic energy and strain can be obtained by fitting the calculated E − V data to the Birch-Murnaghan EOS. The EV data used the Birch-Murnaghan EOS to get the bulk modulus and equilibrium vol-ume, thereafter the P V data was obtained. The strain was obtained with different ranges from −10 GPa (compression) to +10 GPa (tension) with an interval of 5 GPa. As a result, the volume of the structure was affected and the Seebeck coefficient was calculated at different volumes.
2.3
The Boltzmann Transport Equation
The Boltzmann Transport Equation (BTE) is a common theoretical equa-tion used in TE materials studies, and has been used for many classical systems. It was extended early on to apply to the transport by quantized particles, particularly by electrons in metals. The BTE is essentially on equation for the phonon distribution function fλ that introduced to
de-scribe allowed energy states. Two factors affect the phonon distribution: diffusion due to the of gradient temperature, ∇T , and scattering arising from allowed processes. In the steady state, (dfλ/dt = 0), the rate of
change in the distribution must vanish. This condition is expressed by the BTE [12]: dfλ dt = ∂fλ ∂t diffusion + ∂fλ ∂t scattering = 0 (4) where ∂fλ ∂t diffusion = −∇T · vλ ∂fλ ∂T (5)
here ∇T is the temperature gradient and vλ is the velocity of phonon mode
λ.
For equilibrium distributions of electrons, the Boltzmann equation is solved by the Fermi–Dirac function:
f0(E) =
1
exp [(E − EF) /kBT ] + 1
(6)
where E is the energy of an electron, kB Boltzmann constant and EF
is called Fermi energy, strongly dependent on carrier concentration and weakly on temperature T .
The electrons will be scattered by the lattice defects, phonons, and grain boundaries in a non-equilibrium state. The distribution is maintained at equilibrium, where at low temperatures, impurities in the material and de-fects will dominate the electron scattering. As temperature increases, the scattering of charge carriers will increase due to the thermal vibration [23]. It is assumed that a relaxation time, τ , can describe the scattering depend-ing on the energy of the carriers, accorddepend-ing to which the scatterdepend-ing process is dominant [23], [24]. It is defined as:
∂fλ ∂t scattering = f0− f τ (7)
here f is the non-equilibrium distribution and f0 is the equilibrium
distri-bution.
Using the relaxation time for the scattering term yields to the departure of the electron distribution, which holds in the absence of fields or temperature gradient. It is assumed that the motion of electrons and phonons can be considered separately apart from the effects of electron-phonon scattering [23]. Thus, the electric field only alters the electron distribution.
2.4
Non-Equilibrium Green’s Functions
Non-equilibrium Green’s functions (NEGF) provide a method for model-ing non-equilibrium quantum transport in open mesoscopic systems that study electrons, spins, and phonons in various condensed matter systems like metals, semiconductors, superconductors [17], [21]. This method is mainly used for ballistic conduction. It enables to describe nonequilibrium extended systems as well as mesoscopic (nanoscopic) systems, which have to be treated like open systems [17]. The formalism of NEGF can be con-veniently used to describe various steady-state and equilibrium situations. The two interested quantities from the NEGF calculation are the current and charge density matrix [14].
The non-equilibrium Green’s function is often calculated since it is eas-ier than solving the Schr¨odinger equation and the whole eigenvalue prob-lems [15]. Most properties of the system can be calculated from Green’s function, where the NEGF can be used with the DFT technique to calculate spin-polarized and quantum transport [22]. The non-equilibrium Green’s function used in density functional theory (NEGF-DFT) is widely used in analyzing non-linear and non-equilibrium quantum transport in molecular electronics. NEGF-DFT is related to the Landaur approach and has proven to be powerful for studying electron transport through nanoscale devices where the device leads and the scattering region are treated automatically equally [22]. The Green’s function defined as:
(E − H)G(E) = I (8) where E is the energy, and H is the Hamiltonian operator of the system, G is the retarded Green’s function and I is the electric current. The Green’s function gives the response of a system to a constant perturbation [15] |vi in the Schr¨odinger equation:
The response of this perturbation is:
(E − H)|ψi = −|vi (10)
|ψi = −G(E)|vi. (11) By adding this perturbation to the Schr¨odinger equation, it turns out that it is easier to calculate the Green’s function [15] than the Schr¨odinger equa-tion.
Many TE calculations of binary cobalt antimonide are based on the WIEN2k package with the semi-classical Boltzmann transport method are used in the relaxation-time approximation [5]. Therefore, the DFT functional the-ory was employed in this study combined with the semi-classical Boltz-mann transport theory and non-equilibrium Green’s function to compare the method that assumed τ to be a constant at 10 fs as well as the chemical potential [5]. From a computational point of view, the NEGF-DFT tech-nique can be implemented in a rather efficient manner to simulate larger systems that require a reasonably sized basis set where a small basis set does not give accurate results. The NEGF-DFT is done along the ~a direc-tion that is done in this work.
In this thesis, NEGF is combined with DFT in order to evaluated the transport distribution (TD) [25] which is important in the calculation of the Seebeck coefficient.
2.5
The Seebeck Coefficient
The Seebeck coefficient is necessary to measure a TE material where the voltage is built up when a small temperature gradient is applied to a ma-terial. The Seebeck coefficient is calculated by the ratio of the potential difference ∆U versus the temperature difference ∆T between the two mea-suring points in contact with each other forming a closed cycle. The equa-tion gives the resulting voltage between two point A, and B :
U = Z T2
T1
(SB(T ) − SA(T )) dT (12)
here, SA and SB are the Seebeck coefficients differing from one material to
another, and temperature T1 and T2 represent the different temperatures
The TD is the kernel of all transport coefficients, which is vital in calcu-lating the Seebeck coefficient, where S is defined as the voltage gradient produced in a sample by a given temperature gradient when the electrical current is zero [25]. The transport coefficients are necessary to determine the Seebeck coefficient, which is calculated numerically by the given equa-tion [25]: σ = e2 Z dE −∂f0 ∂E Ξ(E) (13) S = ekB σ Z dE −∂f0 ∂E Ξ(E)E − EF kBT (14)
here, f0 is the Fermi Dirac distribution, E is the energy, EF is Fermi
energy, σ electrical conductivity, kB is the Boltzmann constant, e is the
electric charge and, Ξ is the transport distribution coefficient.
2.6
Density Functional Theory Based Molecular
Dy-namics
Density functional theory based on molecular dynamics (DFT-MD) is a way to combine molecular dynamics and density functional theory, which is a rapidly evolving and growing technique for the realistic simulation of complex systems. DFT-MD is also constrained by the density functional theory DFT approximations it can use, which are necessarily a compro-mise between efficiency and accuracy [58]. The molecular dynamics relies on the semi-empirical adequate potential energy of a system expressed as a function of the nuclear coordinates, which approximate quantum effects, while the physical potential in DFT-MD are real.
In a DFT-MD calculation, the forces are obtained ”on the fly” from the electronic structure calculation and permit the chemical bond breaking and account for electronic polarization effects [27]. The DFT-MD calculation assumes the perfect form (ideal) of the system consisting of N nuclei and Ne
electrons, that the Born–Oppenheimer approximation can be used which the nuclei dynamics can be treated classically on the ground-state electronic surface. The total Hamiltonian is [27]:
here the terms are the electronic kinetic energy, the electron–electron re-pulsion, the electron–nuclear attraction, the nuclear kinetic energy and the nuclear–nuclear repulsion, respectively. The classical dynamics of the nu-clei is given by an equation of motion [27]:
MIR¨I = −∇I[E0(R) + VN N(R)] (16)
here, MI is the nuclear mass and E0(R) is the corresponding ground-state
energy eigenvalue at the nuclear configuration R. On the other hand, the ground-state electronic problem cannot be solved precisely. Thus, approx-imate electronic structure methods are needed.
The popular electronic structure method that is currently used in DFT-MD simulations is the KS density functional theory. The KS can provide accurate potential energy for the systems of interest since there are typically 104− 106 time steps per DFT-MD trajectory [28].
In this study, DFT-MD method was employed to tackle the underlying ox-idation mechanisms of CoSb3. The DFT-MD simulations were performed
with the OpenMX software package due to its computation efficiency com-pared to other DFT codes [29].
In DFT-MD, the full structural optimization was carried out for each con-figuration with the convergence criterion for the KS orbitals where the energy cut-off was set to be 150 Ry, and BZ sampling at 1× 1 × 1 K-grids. The system must be large enough to minimize the self-interaction between the atoms. The DFT-MD was simulated of 128 atoms, 2 × 2 × 1 CoSb3
bulk supercell. Atomic structures were represented using the VESTA pack-age [53]. DFT-MD simulations were carried out at approximately 600 K (NVT-vs) velocity scaling thermostat and a canonical ensemble (number of particles N , volume V , temperature T ).
The DFT-MD simulations for CoSb3 were done at 10 000 steps (each 1
fs), and convergence criteria were set at 10−4 Hartree to speed up the simulations and minimize the computational time. The MD step in the oxidation study started with one O2 molecule that was randomly placed at
a distance of about 2 ˚A from the CoSb3 surface. Thereafter 10 O molecules
were used in the oxidation study. The underside layer was frozen to mimic the infinitive bulk where the upper surface was free to move to allow the oxidation to happen. The results shown were visualized using the software visual molecular dynamics VESTA.
3
Results and Discussion
3.1
Ideal CoSb
3The total energy as a function of the volume CoSb3 structure is presented
in figure 2a. This figure shows the data obtained from the simulations us-ing DFT-PBE. Moreover, the 0 K properties of the CoSb3 structure from
this method are listed in table 1 with previous theoretical and experimental values. The value of the lattice constant predicted by this calculation is a = 9.134 ˚A that is in good agreement with the literature’s values where different experimental values are presented which found a lattice constant a = 9.034 and 9.038 ˚A [36], [37]. The discrepancy is 1.09 % and 1.05 %, re-spectively. However, this is quite a satisfactory result since the deviation is less than the largest relative deviation in the lattice constants of 2 % as dis-cussed by Paier et al [48]. The theoretical lattice parameter was compared with the PBE value. For example, M. R˚asander and M. A. Moram [35] determined a value of a = 9.174 ˚A by using the RPBE method, and they found a value of a = 8.910 ˚A obtained by the local-density approximation (LDA) method. The discrepancy is 0.43 % with respect to the first method value and 2.45 % for the second. The RPBE method is a revised version of PBE that improves atomization energies but underestimates cohesive energies of solids, which explains the convergence of the results of lattice constants of CoSb3 between BPE and RPBE.
The bulk modulus value obtained from this study was also compared with other published results. Considering the three experimental data [35], [36], [37] cited in table 1, the bulk modulus ranges of 81 − 85 GPa. It is close to the theoretical data in this study. The numerical accuracy of the compu-tations is in the same region as the experimental result and shows a good agreement. The error is less than the largest relative error in the bulk mod-ulus for some semiconductors, which is 10 % discussed by Paier et al [48]. Moreover, the RPBE method shows a perfect agreement to the theoreti-cal result in this study, where the discrepancy is 0 %. The lotheoreti-cal-density approximation is valid only for systems with slowly varying densities, and it is overestimated cohesive energies of metals. The theoretical calculation of the PBE functional in this study reveals more accurate results than the LDA functional, this conclusion is based on the comparison of the bulk modulus between the experimental data by Matsui et al. [36] and the cal-culated LDA functional which shows a deviation of 23.80 %. However, it
shows a less deviation, only 6.32 %, when compared to this study that used PBE functional.
(a) (b)
Figure 2: a) Total energy as a function of volume for CoSb3. b) Pressure as
a function of volume, negative sign (compression), positive sign (tension). The data points are fitted by polynomial to guide the eyes.
Table 1: Comparison of the value of the lattice parameter and the bulk modulus obtained in this study with other reported in the literature.
Method Lattice parameter [˚A] Bulk modulus [GPa] Reference
PBE 9.134 79 This work
RPBE 9.174 79 [35]
LDA 8.910 104 [35]
Experimental 9.034 84 [36] Experimental 9.038 81 [37]
Experimental - 85 [5]
• LDA: Local density approximation. Even though the LDA is a very simple approximation, it has been found to give good results for many material properties. However, this accuracy level is not sufficient for the application mentioned in this study [35].
The Seebeck coefficient is a transport parameter of utmost importance in TE. There are not many studies examining the influence of the strain on the Seebeck coefficient for pure cobalt antimonide, but some articles study the properties of the Seebeck coefficient under doping with other materials. The calculations implemented in this method are based on the NEGF con-dition and pure binary CoSb3. Figure 3 shows the Seebeck coefficient as
a function of strain and the EF. The theoretical calculation in this study
is applied at 300 K, where the electronic structure is normally calculated at 0 K. In order to examine the influence of the temperature variation, the offset of the EF was considered. The EF used at two different offsets to
check how that will affect the Seebeck coefficient.
Before analyzing the Seebeck coefficient’s behavior as a function of strains, the Seebeck coefficient was examined at equilibrium. However, the value obtained is −29 µVK−1 deviates from the obtained results for the same compound in the theoretical and experimental study [40], [41], [42], [43]. The inset theoretical and experimental data in figure 3 shows the Seebeck coefficient exhibit scattering in a large range from −200 to 200 µVK−1 so, the calculated data obtained in this method is sound. This variation could be due to the different methods assumed constant relaxation time and chemical potential where that did not adopt in this method.
Figure 3 also presents the Seebeck coefficient as a function of strain. The Seebeck coefficient was studied at different offsets of EF; at original EF
obtained from the simulation, +0.1 and −0.1 eV offsets. For simplicity, the non-offset Fermi energy will discussed first. The strain at −10 GPa shows that the absolute value of the Seebeck coefficient increases to 34 µVK−1 and that agrees with the theoretical behavior [5] where the Seebeck
coefficient increase under compression. Moreover, an arrow symbol in the figure 3 denotes an increase of the absolute value of the Seebeck coeffi-cient with increasing compression. These trends are consistent with the theoretical data of Hu et al [5]. This absolute value increases due to the electronic structure, where the electronic structure was affected by stress. The strain enlarges the bandgap, suppresses the density of state near the valence band edge, and fosters the band convergence between the valley bands and therefore the conduction band minimum [5].
Figure 3: Calculated Seebeck coefficient as a function of pressure from −10 GPa (compression) and +10 GPa (tension) at 300 K. The colored lines are presented to visualize the trend of the Seebeck coefficient by offsets of the Fermi energy (EF). (The arrow indicating the trend of Seebeck coefficient
under compression from(Hu et al [5])).
Furthermore, the Seebeck coefficient increases significantly with increasing tension. For +10 GPa, the Seebeck coefficient increases to +894 µVK−1. As shown in figure 3, the Seebeck coefficients increased with the offset of the EF to give a higher absolute value of the Seebeck coefficient at +0.1
eV under compression. Moreover, the Seebeck coefficient became higher at positive offset under tension. The influence of the temperature (offset of the Fermi energy) does not influence the trends in the Seebeck coefficient.
According to the various effects on the Seebeck coefficient, the electron den-sity maps were obtained to clarify how the Seebeck coefficient is affected through the electronic structure. The behavior of the electron density be-tween Sb and Co was analyzed in the case of compression, equilibrium, and tension on the (110) plane. The 2D DFT electron density maps depicted in figure 4(a-c) are drawn in the range 0 to 0.3 e/˚A3.
Figure 4: A (110) slice illustrates electron density distributions map for ideal CoSb3 obtained from density functional theory simulation at 300 K.
By comparing the results of the electron distribution at a compression of −10 GPa in figure 4a and at equilibrium 0 GPa in figure 4b, the differ-ence of the electronic structure between Sb and Co atoms can clearly be observed. The electron density distribution at compression is increased between the Sb and Co atoms due to the enlarged electron density clouds around the Sb and Co atoms in CoSb3, which perfectly implies the presence
of higher number of electrons in CoSb3, indicating high concentration of
localized charges. However, the difference in the electron distribution can be observed at the tension of +10 GPa, where the density electron distri-bution is decreased between Sb and Co atoms in figure 4c. The electron density clouds around the Sb and Co atoms are shrunken in CoSb3, which
have a lower number of electrons in CoSb3, indicating lower concentration
of localized charges. The increasing of the electron density distribution at compression and the decreasing at tension show consistency with the results obtained by Hu et al [5].The behavior of the electronic structure ex-plains this difference in the Seebeck coefficient at tension, which could be related to changes in the bandgap. According to the significant influence of the tension on the Seebeck coefficient, it might be interesting to explore how the tension affects the bandgap in future work.
3.2
Real CoSb
3The material properties are determined by the crystal structure, atomic constituents, spatial arrangement, and how the spatial arrangement varies in the material (micro-structure). The defect can occur at any of those lev-els, as well as it can results from oxidation or doping of the materials. How-ever, some defects can cause a vital change in the properties, for instant, point defects affect important physical properties such as diffusivity [45] and electrical conductivity [46]. The defects can affect the scattering of the phonon which in turn lower the thermal conductivity [50]. Another exam-ple of the defects is doping of semiconductors that increases the electrical conductivity and precipitation hardening of metallic alloys [47].
Point defects are zero-dimensional defects in which the atomic arrangement deviates from the crystal lattice. Typically, point defects could be vacan-cies, interstitial atoms, or substitution atoms. Oxidation, in general, can enhance the effect of the point defects on the transport properties of TE
materials [49]. The atomic mechanisms under the oxidation process can explain how the CoSb3 is affected. In the present thesis, the oxidation was
investigated using DFT-MD calculations. The comprehensive aim is to see how oxidation affected the point defects on the CoSb3 and how these
de-fects affected the Seebeck coefficient.
Figure 5: Atomic processes occurring during oxidation of CoSb3 at 600
K, a) represent the initial structure of CoSb3 with a molecule of oxygen
above the (001) surface at 0 fs, b) oxidation of 5 O2 in CoSb3 at 5000 fs,
c) oxidation of 10 O2 in CoSb3 at 10 000 fs.
Cobalt antimonide oxidation is presented in figure 5. The oxidation occurs in several stages, which start from 0 to 10 000 fs. Figure 5a shows the atomic model of the early stage of the oxidation, where only one oxygen molecule is presented above the CoSb3 surface (001). Figure 5b shows the
defect of the structure of CoSb3 at 5000 fs, where the amount of oxygen is 5
O2at 600 K. As a result the defects cause Sb vacancies and oxygen
intersti-tial, which is illustrated in figure 6 and 7. It is noticeable that the surface (layer 1) is more affected than the other layers where the oxygen molecules are diffused and spread out in CoSb3. Oxygen inward diffusion and Sb–O
exchange seem to give rise to surface amorphization. That means there is a clear tendency for oxygen to cluster horizontally in the surface of CoSb3.
Figure 5c shows the final stage of oxidation at 10 000 fs, where the oxygen atoms interact inside the supercell CoSb3 at an average temperature of 600
K. The elevated amount of oxygen increases the surface amorphization due to enlarged structure defects. Moreover, the increased amount of oxygen increases the Sb and Co dissociation by breaking the bonds between then as well as increases the oxygen diffusion in CoSb3. During the oxidation, the
CoSb3 structure. Thus the bonds inside the structure will start to break
apart which in turn causes the Sb vacancies. This diffusion of oxygen atoms increases mainly in the areas where the vacancy defects took place in the surface layer of CoSb3. It is noticeable that at this final stage, the
oxygen atoms penetrate the CoSb3 sub-layers (layer 2) and (layer 3). The
vacancy defect shows no effect on Co comparing to the great effect on Sb in the CoSb3 structure. In the last stage of oxidation, the inward oxygen
diffusion is considerably slow but active, whereby more Sb–O bonds are formed than Co–O which suggests higher reactivity of Sb.
Figure 6: Schematic of oxygen diffusion through CoSb3at 600 K. Individual
oxygen atom penetrating through Co and Sb layers are highlighted.
Figure 6 shows the dissociative chemisorption of O2 on the CoSb3 surface
(001). Chemisorption is the dissociation of molecules such as oxygen on the metal surfaces. The oxygen molecules are adsorbed into the surface then diffused across the surface to the chemisorption sites. The chemisorption process breaks the bonds in favor of forming new ionic bonds in the surface. DFT-MD is used to investigate the oxidation process of CoSb3 at
temper-ature 600 K. The oxygen atoms move from one interstitial site to another neighboring interstitial site. Thus, oxygen undergoes the inward intersti-tial diffusion between Co and Sb. The oxygen take approximately 1000 fs to go under the surface and occupy interstitial sites inside the CoSb3. Sb
vacancies occur every 1000 fs due to the interstitial oxygens that enter the CoSb3 structure and cause the bonds between Sb and Co. It is noticed
that the dissociative chemisorption process shows that the oxygen enters the CoSb3 before vacancies occur, on the other hand, the longer time of
Figure 7: Schematic of CoSb3 structure at 600 K where Sb atoms are
displaced from their original place and leave vacancies.
Figure 7 represents the Schematic of Sb vacancies in the CoSb3 structure
at 600 K, where the displaced Sb atoms (in brown) have become vacant in the structure of CoSb3. DFT-MD analysis of the CoSb3 indicates that
only antimony vacancies is presented in the structure as shown in figure 7. As acknowledgment, this kind of defect has been discovered in previous studies. For example, losing Sb atoms due to evaporation or forming a secondary phase with the filler elements during material synthesis [54]. Creating Mg vacancies in Mg2−δSi0.4Sn0.6 during synthesis or in subsequent
heat treatments where Mg is oxidized or evaporated [51]. Also a similar point defect was observed in the heat-treated Ag1−xInTe2 sample, which
showed the Ag vacancy concentration increases [52].
In order to clarify how the defects influence the Seebeck coefficient at the CoSb3 structure, DFT calculation were conducted with Sb vacancies
de-fects. The vacancies are taken from the oxidation part where two Sb atoms are removed out of 24 atoms from the ideal structure, and one oxygen atom interacts inside the structure as shown in figure 8. The position of the Sb vacancy was randomly chosen from the oxidation simulation.
Figure 8: The unit cell shows the real CoSb3 structure. (Two Sb were
removed and one interstitial oxygen entered the structure).
Figure 9 shows the Seebeck coefficient of the real and ideal CoSb3 structure
under strain. As shown in figure 9, the Sb vacancy defects profoundly alter the Seebeck coefficient compared to the ideal structure of CoSb3. The Sb
vacancies raises the Seebeck Coefficient at equilibrium to 151 µVK−1. A previous study found that vacancies can increase the Seebeck coefficient of individual Carbon nanotubes by up to four times, due to the selective sup-pression of the transmission function at energy levels corresponding to Van Hove singularities [55]. However, with increasing strain at −10 GPa, the Seebeck coefficient shifted from negative to positive, which is considered as changing the type of conductivity from n-type to p-type [56].
Furthermore, for compression at −5 GPa and tension at +5 GPa, the See-beck Coefficient shows an inverse sign compared to the ideal CoSb3. The
calculated data exhibits a deviation of Seebeck Coefficient at tension +10 GPa where it shows a downward trend to −270 µVK−1. This unexpected behavior contradicts with the ideal structure at tension, there is a clear sig-nificance of the point defect that causes this difference. This implies there is a relation between the vacancies and the Seebeck coefficient which is illustrated by the reflection of the Seebeck Coefficient of real CoSb3
struc-ture at tension.
Many reasons may affect the Seebeck coefficient, and among these reasons is the electron density distribution between the Sb and Co atoms. The electron distribution can also be affected by several factors such as strain or vacancies.
Figure 9: Calculated Seebeck coefficient at ideal and real structure of CoSb3
structure as a function of pressure from −10 GPa (compression) and +10 GPa (tension) at 300 K. The fitted line are presented to visualize the trend of the Seebeck coefficient and guide the eyes.
Figure 10 represents the electron density distribution map for the (110) plane of real CoSb3. it shows the difference in electron density
distribu-tion between Sb and Co atoms under strain and vacancy defects. Under compression −10 GPa, the electron distribution increases between Sb and Co atoms while compared to equilibrium, which causes enlargement of the electron density clouds around the Sb and Co atoms in CoSb3 based on
high concentration of localized charges inside the structure. However, the electron distribution between Co and Sb atoms at tension decreases com-pared to equilibrium and compression. This means that the electron density clouds around the Sb and Co atoms are reduced in CoSb3 based on the low
concentration of localized charges inside the structure.
At equilibrium, the electron density clouds around the Sb and Co atoms are enlarged in the real CoSb3 compared to the ideal structure, which
GPa, the electron density distribution in the real structure is increased between Sb and Co atoms comparing to the ideal structure. This increase in the localized charges results in increasing of the Seebeck coefficient. The direction of the Seebeck coefficient is unexpected at the real structure at tension +10 GPa, where the Seebeck coefficient decreased from 894 µVK−1 in the ideal structure to −270 µVK−1 in the real structure. By
inspection of the ellipticities at the (110) plane of CoSb3 it becomes clear
that the electron density distribution between Sb and Co exhibit completely different behavior at tension +10 GPa. It is noticeable that there is a decrease in electron density distribution between Sb and Co, where the orbital overlap shows a decrease between the Sb and Co, indicating a lower concentration of localized charges.
The electron density distribution in the CoSb3 with Sb vacancies and
un-der strain is consistent with the electron density distribution unun-der strain in ideal CoSb3. However, the Seebeck coefficient did not show this
con-sistency. The vacancy defects affect the electron distribution between Co and Sb atoms and the interaction of electron clouds of these atoms, which in turn affects the Seebeck coefficient. The decrease of the Seebeck coef-ficient at tension is still unknown, it may be related to interstitial oxygen that enter the real structure during oxidation. Therefore, further studies are performed on the electron density distribution between oxygen and Sb atoms to see how it affects the Seebeck coefficient.
Figure 10: A (110) slice illustrates electron density distributions map for real CoSb3 obtained from density functional theory simulation at 300K.
Figure 11: A (224) slice illustrates electron density distributions map for real CoSb3 to obtain the electron distribution between oxygen and
The electron density distribution map presents the plane (224) between oxygen and Sb atoms in figure 11. This figure shows the real structure at tension and equilibrium. The electron density map shows high electron distribution between the O and Sb atoms at equilibrium. The increased strain from compression all the way to tension in real CoSb3 causes the
notable depletion of orbital overlap. This is not consistent with the See-beck coefficient going from 18 to −270 µVK−1. In the presence of oxygen (real CoSb3), the electronic density distribution between oxygen and Sb at
equilibrium is higher than at the tension +10 GPa. The electronic density distribution is the same, but the Seebeck coefficient at tension is different compared to the ideal CoSb3 structure. The trend is changed (reverse).
Thus, the analyzed electronic structure is made to explain this depletion at tension which is not clear at the moment. The question is why this reverse of the trend occurs? This could be an ionic effect due to the transfer of the charges occurring. It could be an effect that could not be captured in this analysis with vacancies. This study was an attempt, and therefore further studies are needed to be conducted.
4
Conclusions
In this thesis, two different CoSb3 structures were considered, an ideal one
(without any defects) and the other is termed real (containing defects). The Seebeck coefficient of the CoSb3 structure was studied for its
corre-lation with two factors: strain and point defects. Based on the strain, CoSb3 at equilibrium shows the Seebeck coefficient is 29 µVK−1, while at
compression and tension, the trends of the Seebeck coefficient are changed. The Seebeck increased from −34 µVK−1at compression to 894 µVK−1at tension, which was unexpected. The theoretical results of the Seebeck-increase need to be validated experimentally and exploited in applications. The electron density between Sb and Co is higher at compression, which shows consistency with the Seebeck coefficient, while this relation between electron density and Seebeck shows a contrast at tension. The effect of EF
was studied, and it was found that the offset of the EF by +0.1 eV can
increase the Seebeck coefficient of CoSb3 at compression and tension.
Based on point defects, the CoSb3 structure was affected by the presence
of oxygen molecules. Oxygen penetrated through both Co and Sb atoms in CoSb3 structure, however, it preferentially interacts with Sb. The
va-cancy defects occur in CoSb3 Structure by oxidation, and it is Sb atoms
are removed from the structure. After the vacancy defects, the Seebeck coefficient showed a considerable increase at equilibrium compared to the ideal structure. The elevation of the Seebeck coefficient was due to the electron density, which was verified by the electron density map that shows an evident rise of the density of the electrons between Sb and Co atoms. As a consequence, the equilibrium Seebeck Coefficient of CoSb3 of 29 µVK−1
was increased to 151 µVK−1 for the real CoSb3. At tension, the Seebeck
coefficient decreased to −270 µVK−1. The trends of the Seebeck coeffi-cient were verified by the electron density map that shows an evident rise in the density of the electrons between Sb and Co atoms at equilibrium and compression but not at tension. The interstitial of the oxygen atoms inside the real CoSb3 structure affected the Seebeck coefficient significantly.
The Seebeck coefficient could be well enhanced at compression and tension in the ideal CoSb3 structure. The Vacancies of Sb could occur because of
the oxidation where the oxygen molecules penetrate in CoSb3 structure.
Enhancing the Seebeck coefficient in the real structure occurred at equi-librium, at compression, but not at tension. The interstitial oxygen could play an essential role in increasing the electron density in which in turn affected the Seebeck coefficient.
5
Future Work
Evaluations of the Seebeck coefficient in the ideal and real structure are performed under strain. During the strain, the Seebeck coefficient was studied at equilibrium, compression, and tension with a step between them. It shows a rare behavior during tension as the Seebeck coefficient increases and shows a good performance. This result must be verified and taken advantage of by this rise. Future studies at the tension of CoSb3 should
include evaluations of the bandgap that will explain electronic structure at tension. Moreover, it is possible to detect the status of the system if it is isotropic or not.
As discussed in a previous section, altering the bandgap in the CoSb3
struc-ture could affect the Seebeck coefficient that affects the figure of merit ZT . The real structure was investigated and showed a good performance for the Seebeck coefficient at equilibrium and compression. The real CoSb3
shows a decreasing and reverse trend of the Seebeck coefficient at tension. Findings regarding the Seebeck coefficient losses indicate that this loss’s underlying effect should be studied further at tension. Efforts should be made in finding the electronic structure layout of these changes at tension.
The vacancies of Sb atoms significantly affect the Seebeck coefficient, but it is still unknown how the Seebeck coefficient will be affected by the location of the vacancies under strain.
TE research is a continuous process, and findings in new high efficient mate-rials in the future will have great significance in establishing a system with distinctly higher net power output. Efforts could also be used to influence the direction of development towards CoSb3 by investigating various effects
found in this research at tension.
6
Acknowledgements
The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at National Supercomputer Centre (NSC) in Link¨oping, Sweden.
I would like to thank and express my sincere appreciation to my advisor Dr. Denis Music who gave me great guidance, support, and encouragement dur-ing my thesis. With his great experience in thermoelectric materials and his valuable instructions I could progressively complete my scientific research. I would also like to thank all the other doctors for their contributions during my two-year master studies.
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Appendix
A Matlab code
In this appendix all Matlab code is given. It consists of several scripts, which cover some interpolation of data described in this paper.
A.1 Energy.m clear all; clc; close all; % CoSb3 for k= 7 * 7 *7. dirpath Out=’C:/Users/user/OneDrive/Desktop/Energy’; a=[8.8; 8.9; 9.0; 9.1; 9.2; 9.3;9.4;9.5].3; E=[-2937.396266213930 ;-2937.461138816272 ;-2937.505699063516 ; -2937.532103764210; -2937.542287283410; -2937.537759344508; -2937.519993243058; -2937.490360210835].*2; p=polyfit(a,E,3); x=linspace(650,900,100); y=polyval(p,x); plot(a,E,’o’); % plot E vs V hold on plot(x,y) ytickformat(’%.1f’) xlabel(’Volume [˚A3]’,’FontSize’,20)
ylabel([’Energy [Ry]’],’FontSize’,20)
set(gca,’FontSize’,17)
fileNameOut = sprintf(’%sE@7.png’,dirpath Out);
saveas(gcf,fileNameOut) % saving the figure of energy
A.2 pressure.m clear all; clc; close all; % CoSb3 for k= 7*7*7. dirpath Out=’C:/Users/user/OneDrive/Desktop/test’; a=[8.8; 8.9; 9.0;9.1; 9.2; 9.3].3; p1=-[11.13;7.22;3.83;0.90;-1.62;-3.79]; p=polyfit(a,p1,3); x=linspace(650,900,100); y=polyval(p,x);
plot(a,p1,’o’); % plot pressure vs volume
hold on
plot(x,y)
xlabel(’Volume [˚A3]’,’FontSize’,20) ylabel([’Pressure [GPa]’],’FontSize’,20)
xtickformat(’%.f’)
set(gca,’FontSize’,17)
fileNameOut = sprintf(’%s/ps@7.png’,dirpath Out);
saveas(gcf,fileNameOut)
% calculation of the fermi dirac
A.3 Fermidirac.m
function f0 = fermidirac ( E,EF,T )
kb = 8.61734310ˆ 5; %Boltzmann constant
f0 = 1 . / ( exp( (EEF) . / ( kb . T) ) +1) ;
end
A.4 Difffermi.m
% Function that make a derivative of fermi energy as a function of Energy
function df0 = difffermi( E,EF,T )
kb = 8.617343 * e−5; %Boltzmann constant
df0 = -exp( (E-EF) . / ( kb . * T) ) . / ( kb . * T. * ( exp( (E-EF) . / ( kb * T) )
+1).ˆ2 ) ;
end
A.5 calculated Sebeek
clear all;
T=300;
FE=-4.299285e+00; % Fermi energy e = 1.60217653 * e−19; %Elementary charge RK=1.38064852 * e−23; inti=trapz(E,TD.*(-difffermi(E,FE,T))) segma=e2 * inti L=trapz(E,TD.*(-difffermi(E,FE,T)).*((E-FE)./(K.*T))) S=(e*RK/segma)*L*106 A.6 Seebeck-graph.m clear all; clc; close all; dirpath Out=’C:/Users/user/OneDrive/Desktop/test’;
% CoSb3 for k= 7*7*7 for ideal structure.
a=[-10 ; -5; 0; 5; 10]; E=[-34.443164202189340;23.489076759414726;-28.975697094635514;28.577476774 079454;8.936458926370140e+02]; % at 0.1 offset: E1=[-16.588657055427250;32.139031124565620;-31.713104920752052;-46.65663467 4468500;1.128204726460391e+03]; % at minus 0.1 offset: E2=[-69.237147244837390;10.246495156191527;-25.653271335449766;
-18.477900050028374;5.647678859594081e+02]; figure(1) p=polyfit(a,E,4); x=linspace(-10,10,100); y=polyval(p,x); figure(1) F1=plot(a,E,’o’,’MarkerFaceColor’,’red’,’DisplayName’,’EF.’); ylim=([-600 1200]) hold on plot(x,y,’red’) hold on q=polyfit(a,E1,4); x=linspace(-10,10,100); f=polyval(q,x); F2=plot(a,E1,’s’,’MarkerFaceColor’,’black’,’DisplayName’,’EF + 0.1 eV.’); hold on plot(x,f,’black’) k=polyfit(a,E2,4); x=linspace(-10,10,100); h=polyval(k,x); hold on
F3=plot(a,E2,’d’,’MarkerFaceColor’,’black’,’DisplayName’,’EF - 0.1 eV.’);
hold on
plot(x,h,’blue’)
hold on
xlabel(’Pressure [GPa]’,’FontSize’,10)
ylabel([’Seebeck Coefficient [10−6V K−1]’],’FontSize’,10) set(gca,’FontSize’,12) F4=plot(0,108,’p’,’MarkerFaceColor’,’m’,’DisplayName’,’Watcharapasorn et al.’); F5=plot(0,185,’¡’,’MarkerFaceColor’,’c’,’DisplayName’,’Nolas et at.’); F6=plot(0,220,’v’,’MarkerFaceColor’,’black’,’DisplayName’,’Caillat et al.’); F7=plot(0,-220,’ ˆ ’,’MarkerFaceColor’,’green’,’DisplayName’,’Kawaharada et al.’); hleg=legend([ F1 F2 F3 F4 F5 F6 F7],’Location’,’north’) hleg.FontName =’Ariel’; hleg.FontSize=11; hleg.FontWeight=’normal’;
fileNameOut = sprintf(’%s/add70.1.png’,dirpath Out);
B IN.dat
In this appendix, there is an example of OpenMX code that using PBE functional. It s contains 32 atoms and their positions for the CoSb3 struc-ture that’s studied under strain in this thesis.
# File Name # CoSb3 System.CurrrentDirectory ./ # default=./ System.Name out level.of.stdout 1 # default=1 (1-3) level.of.fileout 1 # default=1 (1-3) DATA.PATH ../DFT DATA19 #
# Definition of Atomic Species # Species.Number 2 <Definition.of.Atomic.Species Co Co6.0S-s2p3d2f1 Co PBE19S Sb Sb7.0-s3p3d3f2 Sb PBE19 Definition.of.Atomic.Species> Atoms Atoms.Number 32 atoms.SpeciesAndCoordinates.Unit Ang <Atoms.SpeciesAndCoordinates 1 Co 1.7290773 1.7465209 1.7379118 7.50000 7.50000 2 Co 6.9163532 2.3603911 2.3568488 7.50000 7.50000 3 Co 2.3509267 6.9352278 2.3575996 7.50000 7.50000 4 Co 6.9629312 6.9760107 2.3383954 7.50000 7.50000 5 Co 6.9571902 6.9597218 6.9535268 7.50000 7.50000 6 Co 2.3471766 6.9162104 6.9509726 7.50000 7.50000 7 Co 6.9258065 2.3404700 6.9647395 7.50000 7.50000 8 Co 2.8992894 2.9250883 2.9793685 7.50000 7.50000 9 Sb 7.7683919 0.0461256 6.1501822 7.50000 7.50000 10 Sb 7.8011757 0.0496620 3.1166753 7.50000 7.50000 11 Sb 1.5122892 0.0803930 6.2778660 7.50000 7.50000 12 Sb 1.4382194 9.0801552 3.2940722 7.50000 7.50000
13 Sb 3.1671973 1.5028160 8.9784257 7.50000 7.50000 14 Sb 6.1662646 1.4994931 0.0456824 7.50000 7.50000 15 Sb 3.1327298 7.8011312 0.0278014 7.50000 7.50000 16 Sb 6.1510008 7.8091645 0.0355348 7.50000 7.50000 17 Sb 6.0901237 4.6774034 1.5323840 7.50000 7.50000 18 Sb 0.0449159 6.2615445 1.4626343 7.50000 7.50000 19 Sb 3.2250547 4.7782692 1.3912568 7.50000 7.50000 20 Sb 9.0781559 3.2914535 1.4468200 7.50000 7.50000 21 Sb 4.7698756 1.3805369 3.2312753 7.50000 7.50000 22 Sb 4.6468157 7.6389132 3.2039032 7.50000 7.50000 23 Sb 7.7331920 6.1010397 4.6411445 7.50000 7.50000 24 Sb 7.6336560 3.1913020 4.6564456 7.50000 7.50000 25 Sb 1.4006326 3.1079496 4.9125208 7.50000 7.50000 26 Sb 1.5407287 6.0275847 4.6523182 7.50000 7.50000 27 Sb 4.6415813 7.6668686 6.1080182 7.50000 7.50000 28 Sb 4.6160257 1.4772460 6.1253721 7.50000 7.50000 29 Sb 0.0327219 6.1840178 7.7765910 7.50000 7.50000 30 Sb 6.1036294 4.6432577 7.7311479 7.50000 7.50000 31 Sb 0.0268645 3.1562119 7.7717392 7.50000 7.50000 32 Sb 3.1488475 4.5352800 7.6143069 7.50000 7.50000 Atoms.SpeciesAndCoordinates> Atoms.UnitVectors.Unit Ang <Atoms.UnitVectors 9.217171717171718 0.000000000000000 0.000000000000000 0.000000000000000 9.217171717171718 0.000000000000000 0.000000000000000 0.000000000000000 9.217171717171718 Atoms.UnitVectors> SCF or Electronic System
scf.XcType GGA-PBE # LDA|LSDA scf.SpinPolarization off # On|Off
scf.SpinOrbit.Coupling off # On|Off, default=off scf.ElectronicTemperature 700.0 # default=300 (K) scf.energycutoff 150.0 # default=150 (Ry) scf.maxIter 150 # default=40
scf.EigenvalueSolver NEGF # Recursion|Cluster|Band|NEGF scf.Ngrid 64 64 64 # n1, n2, n3
scf.Kgrid 7 7 7 # means 4x4x4 scf.dftD on # on|off, default=off version.dftD 3 # 2|3, default=2 Grimme
